Weyl group multiple Dirichlet series constructed from quadratic characters

We construct multiple Dirichlet series in several complex variables whose coefficients involve quadratic residue symbols. The series are shown to have an analytic continuation and satisfy a certain group of functional equations. These are the first examples of an infinite collection of unstable Weyl group multiple Dirichlet series in greater than two variables.Comment: incorporated referee's comment

A Yang-Baxter equation for metaplectic ice

We will give new applications of quantum groups to the study of spherical Whittaker functions on the metaplectic $n$-fold cover of $GL(r,F)$, where $F$ is a nonarchimedean local field. Earlier Brubaker, Bump, Friedberg, Chinta and Gunnells had shown that these Whittaker functions can be identified with the partition functions of statistical mechanical systems. They postulated that a Yang-Baxter equation underlies the properties of these Whittaker functions. We confirm this, and identify the corresponding Yang-Baxter equation with that of the quantum affine Lie superalgebra $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(1|n))$, modified by Drinfeld twisting to introduce Gauss sums. (The deformation parameter $v$ is specialized to the inverse of the residue field cardinality.) For principal series representations of metaplectic groups, the Whittaker models are not unique. The scattering matrix for the standard intertwining operators is vector valued. For a simple reflection, it was computed by Kazhdan and Patterson, who applied it to generalized theta series. We will show that the scattering matrix on the space of Whittaker functions for a simple reflection coincides with the twisted $R$-matrix of the quantum group $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(n))$. This is a piece of the twisted $R$-matrix for $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(1|n))$, mentioned above

Hecke Modules from Metaplectic Ice

We present a new framework for a broad class of affine Hecke algebra modules, and show that such modules arise in a number of settings involving representations of $p$-adic groups and $R$-matrices for quantum groups. Instances of such modules arise from (possibly non-unique) functionals on $p$-adic groups and their metaplectic covers, such as the Whittaker functionals. As a byproduct, we obtain new, algebraic proofs of a number of results concerning metaplectic Whittaker functions. These are thus expressed in terms of metaplectic versions of Demazure operators, which are built out of $R$-matrices of quantum groups depending on the cover degree and associated root system

The structures of standard (g,K)-modules of SL(3,R)

We describe explicitely the structures of standard $(g,K)$-modules of $SL(3,R)$.Comment: 22 page

Geometric non-vanishing

We consider $L$-functions attached to representations of the Galois group of the function field of a curve over a finite field. Under mild tameness hypotheses, we prove non-vanishing results for twists of these $L$-functions by characters of order prime to the characteristic of the ground field and by certain representations with solvable image. We also allow local restrictions on the twisting representation at finitely many places. Our methods are geometric, and include the Riemann-Roch theorem, the cohomological interpretation of $L$-functions, and some monodromy calculations of Katz. As an application, we prove a result which allows one to deduce the conjecture of Birch and Swinnerton-Dyer for non-isotrivial elliptic curves over function fields whose $L$-function vanishes to order at most 1 from a suitable Gross-Zagier formula.Comment: 46 pages. New version corrects minor errors. To appear in Inventiones Mat